Moduli of Riemann Surfaces, Real Algebraic Curves, and Their by S. M. Natanzon

By S. M. Natanzon

The gap of all Riemann surfaces (the so-called moduli area) performs an incredible function in algebraic geometry and its functions to quantum box thought. the current booklet is dedicated to the examine of topological houses of this house and of comparable moduli areas, equivalent to the distance of genuine algebraic curves, the gap of mappings, and in addition superanalogs of these types of areas. The e-book can be utilized by way of researchers and graduate scholars operating in algebraic geometry, topology, and mathematical physics.

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R + h)}. 9. TOPOLOGICAL CLASSIFICATION OF SYSTEMS OF ARF FUNCTIONS 45 Then the equation to(vi) = a/(v[) implies co(si) = u/(s'), u>(U) = Be­ sides, (Sj, Sj) = (ii> ij) = (®j, <5j) tj) 0, (&i>tj) (®j> tj) &ij) And SO W(V) and W(V') can be extended to symplectic bases {flj, b% (i = 1,.. 1, it is easy to show that the complementary vectors can be chosen in such a way that u>(at) = o/(aQ and u(bi) = ^'(K) for all i. The correspondence a* 1—>a[, bi 1—>b'{ generates an isomorphism : H —►H with the desired properties.

1. 1 applied to the set {C\ , . . , Cn-2, Cn- i , Cn}. □ 2. Now we associate to a lifting of a Fuchsian group T C Aut2(A) U Auti(A) a function •H i(A /r,Z2) —>Z2, where Z2 = Z/2Z = {0,1}. Let P = A /T and let T* C SL(2, R) be a subgroup such th at J induces an isomorphism J : F* —>T. For a given point q € A, an uniformization A —►P induces an isomorphism $ q: T —>7Ti(P,p), where p = $(g) (see Section 2). Construct a function w: n\(P,p) —»Z2 by setting “ (c) \i ¡ M . ): iii( P ,Z 2) x P 2(P, Z2) —>Z2 the intersection form in Hi(P, Z2).

Modp —>Modp/. 1, the element 0 = 4>aiP~l(t>a' generates an isomorphism 0: Mod -»M od . 2 allows us to write Mod instead of Mod^. Each element h € 7 generates the automorphism d 1-» hdh~l of 7 , which belongs to Mod. Denote by IMod C Mod the subgroup of all such diffeomorphisms and set Mod = Mod/IMod. Similarly, each element h € ir (P ^ ,p ) generates the automorphism d i—> hdh~ 1 of 7 ri(P ^ ,p ), which belongs to Modp. We denote by IModp C Modp . — g, — —--0 the subgroup of all such diffeomorphisms and set Mod£ = Modp /IModp .

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